Apparatus and method for synchronizing sampling frequency and/or carrier frequency in a multi-carrier communication system

ABSTRACT

The present invention provides an apparatus and method for synchronizing sampling frequencies of a receiver and a transmitter of a multi-carrier communication system. The receiver includes an estimator for estimating a frequency offset by employing an additional angle rotation of a received signal in frequency domain. The apparatus includes a compensation loop filter for generating a first output in response to a frequency offset compensation, an adder for adding the estimated frequency offset and the first output to generate a second output, and a loop filter for generating the frequency offset compensation according to the second output. The method repeatedly applies the apparatus to generate the frequency offset compensation, and then feeds it back to an oscillator to compensate the sampling frequency of the receiver. The apparatus and method can also be applied to a communication system with a carrier frequency offset.

BACKGROUND OF THE INVENTION

[0001] (a). Field of the Invention

[0002] The present invention relates in general to a multi-carriercommunication system, and more particularly to a compensation apparatusand method for sampling and carrier frequency synchronization in amulti-carrier communication system.

[0003] (b). Description of the Prior Arts

[0004] Multi-carrier modulation is one of wideband technologies andincreasingly important in recent years. FIG. 1 is a block diagram of acommunication system 100 using multi-carrier modulation. Thecommunication system 100 employs a set of N-point inverse fast Fouriertransform (IFFT) 102 in the transmitter and the fast Fourier transform(FFT) 111 in the receiver to transceive data. A channel 106 is dividedinto N sub-channels, and signals of one sub-channel are orthogonal tothose of any other sub-channels. Thus, the data transmitted in thesub-channels would not interfere each other, and inter-channelinterference (ICI) can be avoided.

[0005] The set of N-point outputting from IFFT is called a symbol. Toavoid inter-symbol interference (ISI) and ICI, it is common to add a“cyclic prefix”(CP) to each symbol, i.e. the last υ samples of eachsymbol are copy and added in the front of the symbol. Therefore, asymbol including (N+υ) samples is transceived each time. The circuits103 in the transmitter and 110 in the receiver of FIG. 1 are used to addand remove cyiclic prefixes respectively.

[0006] However, the conventional communication system 100 of FIG. 1suffers the disadvantage of frequency offset. An analog-to-digitalconverter (ADC) 107 is used to sample signals with a sampling frequency{circumflex over (f)}_(s) in the receiver. However, the samplingfrequency f_(s) of the digital-to-analog converter (DAC) 105 in thetransmitter is inconsistent with the sampling frequency {circumflex over(f)}_(s) at the receiver. Therefore, ICI is generated in the output ofthe FFT circuit 111, and a symbol timing error is also increased overtime. The performance of the communication system is thus degradedseriously.

[0007] The conventional approach to overcome the above-disclosed problemis to utilize the delay-rotor property of the communication system 100.FIG. 2 is a diagram illustrating the delay-rotor property of theconventional communication system. Since the sampling frequency of thetransmitter and that of the receiver are different, i.e.f_(s)−{circumflex over (f)}_(s)=df_(s)≠0, the symbol timing error willbe accumulated. This timing error would cause the output signal of theFFT circuit 111 an additional angle rotation in the frequency domain.This rotation angle would also be increased over time, as shown in FIG.2.

[0008] By utilizing the delay-rotor property, the communication system100 may reserve a specific sub-channel for transmitting a given signalcalled pilot tone. When the receiver detects an additional anglerotation of the pilot tone in the frequency domain, the symbol timingerror resulted from the sampling frequency offset can be estimated. FIG.3 is a block diagram of a conventional architecture for dealing with thesampling frequency offset. As shown in FIG. 3, after the circuit 301selects the pilot tone, the angle estimator 302 estimates the angle offrequency-domain signals and passes it to the first difference device303(i.e. 1−z⁻¹) to calculate the additional angle rotation. Next, afterbeing provided into the loop filter H(z) 30.4 and then the accumulator305 $\left( {i.e.\quad \frac{1}{1 - z^{- 1}}} \right),$

[0009] the angle rotation is fed back to a voltage-controlled crystaloscillator (VCXO) 306 to compensate the sampling frequency offset of thereceiver.

[0010] However, the relationship between the sampling frequency offsetand the rotation angle is not clear, it is difficult to design the loopfilter 304. Besides, it takes a very long time for the feedback loop ofFIG. 3 to track the sampling frequency offset. If the sampling frequencyis time-variant, then the tracking performance would be impacted. Evenworse that the feedback loop cannot track the sampling frequency offset.

[0011] In view of this, the present invention provides an apparatus andassociated method that can track and compensate the sampling frequencyoffset more quickly, and tracking delay resulted from the feedback loopdelay is also decreased compared to the conventional method. Meanwhile,the system can still maintain the low complexity as the conventionalsystem.

SUMMARY OF THE INVENTION

[0012] The first objective of the present invention is to provide anapparatus for synchronizing frequencies of a receiver and a transmitterof a multi-carrier communication system, The receiver includes anestimator for estimating a frequency offset by using an additional anglerotation of a received signal in frequency domain. The apparatusincludes a compensation loop filter for generating a first output inresponse to a frequency offset compensation. The apparatus also includesan adder for adding the estimated frequency offset and the first outputfrom the estimator and the compensation loop filter respectively togenerate a second output. The apparatus also includes a loop filter forgenerating the frequency offset compensation according to the secondoutput. The frequency offset compensation is fed back to an oscillatorto compensate the sampling frequency of the receiver, where theoscillator is used to generate the sampling frequency of the receiver.

[0013] The second objective of the present invention is to provide amethod for sampling frequency synchronization by using the aboveapparatus. The core concept of the method is that the frequency offsetestimated by the estimator can be considered as an outcome of thedifference between frequency offsets of the transmitter and the receiverpassing through a system response filter g(z). Thus, if the frequencyoffset compensation is passed through the compensation loop filter witha similar system response g′(z), and the output of the filter is addedto the outcome for compensation, then, the sum is fed back to theoscillator of the receiver. The above process is repeated and thesampling frequency of the transmitter can be tracked step by step.

[0014] The third objective of the present invention is to apply theapparatus and method mentioned above to compensate carrier frequencyoffset. Carrier frequency is used to modulate baseband signals outputtedby IFFT to a higher frequency band. When receiving carrier-modulatedsignals, the receiver demodulates them to baseband signals and then tothe original transmitted signals by FFT. However, since carrierfrequencies of the transmitter and the receiver are not the same, aproblem similar to that of sampling frequency offset is generated. Byapplying the above apparatus and method of the present invention, thecarrier frequency offset can also be tracked and compensated quickly.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015]FIG. 1 is a block diagram of a communication system usingmulti-carrier modulation.

[0016]FIG. 2 is a diagram illustrating the delay-rotor property of theconventional communication system.

[0017]FIG. 3 is a block diagram of a conventional architecture fordealing with the sampling frequency offset.

[0018]FIG. 4 is a diagram showing the relation between the frequencyoffset and the angle rotation of the output of k-th sub-channel from theFFT circuit.

[0019]FIG. 5A is a block diagram showing the behavior of a multi-carriersystem with a sampling frequency offset.

[0020]FIG. 5B is a simplified diagram of FIG. 5A.

[0021]FIG. 5C is a further simplified diagram of FIG. 5B.

[0022]FIG. 6A is a block diagram of the apparatus used in amulti-carrier communication system according to the present invention.

[0023]FIG. 7 is a flow chart of the method according to the presentinvention.

[0024]FIG. 8 is a block diagram showing an application of the behaviormodel of FIG. 6B.

[0025]FIG. 9 is a block diagram showing that a multi-carriercommunication system employs a carrier to transmit signals.

[0026]FIG. 10A and 10B are block diagrams of a behavior model simplifiedfrom the communication system of FIG. 9.

[0027]FIG. 11A is a block diagram showing an apparatus used in thesystem of FIG. 9 according to a preferred embodiment of the presentinvention.

[0028]FIG. 11B is a block diagram showing an apparatus used in thesystem of FIG. 9 according to a preferred embodiment of the presentinvention.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

[0029] It is assumed that the sampling frequency of the transmitter andthat of the receiver of the multi-carrier system 100 in FIG. 1 change soslightly that these sampling frequencies can be considered as constantduring transceiving the same symbol. Therefore, the differencef_(s)(m)−{circumflex over (f)}_(s)(m)=df_(s)(m) can also be regarded asa constant, sphere m is the index of the symbol number. However,df_(s)(m) may be different for different symbols. When df_(s)(m) is notzero, the following effects will occur:

[0030] (1) “constellation distortion” is generated in the output of theFFT circuit 111. Without sampling frequency offset, the n-th time-domainsampling signal S_(m) ^(k)(n) of the k-th sub-channel of the m-threceived symbol given by:

S _(m) ^(k)(n)=a _(m) ^(k) ·H _(k) ·e ^(jw) ^(_(k)) ^(n) _(s)n=0˜N−1  (1-1)

[0031] where a_(m) ^(k) is the transmitted signal of the k-thsub-channel of the m-th symbol, H_(k) is the channel frequency responseof the k-th sub-channel, and _(Wk)=2πk/N is the carrier frequency of thek-th sub-channel.

[0032] Since there is an offset between the sampling frequency{circumflex over (f)}_(s)(m) of the receiver and the sampling frequencyf_(s)(m) of the transmitter, the actual signals sampled by the ADC 107of the receiver are different from the signals shown in Equation (1-1).The sampling frequency offset Δ is defined as df_(s)/f_(s), and thus{circumflex over (f)}_(s)=f_(s)(1−Δ). If S_(m) ^(k)′(n) is used torepresent the n-th time-domain sampling signal of the k-th sub-channelof the m-th symbol actually received by the receiver, then

S _(m) ^(k)(n)=a _(m) ^(k) ·H _(k) ·e ^(jw) ^(_(k)) ^(′n) _(s) n=0˜N−1

[0033] where$w_{k}^{\prime} = {{2\quad \pi \quad \frac{k}{N}\quad \frac{f_{s}}{{\hat{f}}_{s}}} = {{2\quad \pi \quad \frac{k}{N}\frac{1}{1 - \Delta}} \approx {2\quad \pi \quad \frac{k}{N}\left( {1 + \Delta} \right)}}}$

[0034] Next, the N time-domain samples of the k-th sub-channel of them-th symbol are provided to the FFT circuit 111, and then Nfrequency-domain outputs as below are generated:

r _(i) =a _(m) ^(k) H _(k)Φ(W _(i) −W _(k) ^(′))_(s) i=0˜N−1

[0035] where${\Phi (w)} = {{\frac{\sin \left( {{Nw}/2} \right)}{\sin \left( {w/2} \right)}^{{- j}\frac{N - 1}{2}w}} = {{R(w)}{\phi (w)}}}$

[0036] Since the N sub-channels transmit signals simultaneously in apractical system, the output signal of the k-th sub-channel of the FFTcirca it 111 is $\begin{matrix}{r_{k} = {\sum\limits_{j = 0}^{N - 1}{a_{i\quad n}^{j}H_{j}{\Phi \left( {w_{k} - w_{j}^{\prime}} \right)}}}} \\{= {{a_{i\quad n}^{k}H_{k}{\Phi \left( {w_{k} - w_{k}^{\prime}} \right)}} + \underset{\underset{{JCJ}\quad {is}\quad {negligible}}{}}{\sum\limits_{\underset{({j \neq k})}{j = 0}}^{N - 1}{a_{i\quad n}^{j}H_{j}{\Phi \left( {w_{k} - w_{j}^{\prime}} \right)}}}}} \\{\approx {a_{i\quad n}^{k}H_{k}\frac{\sin \left( {\pi \quad k\quad \Delta} \right)}{\sin \left( {\pi \quad k\quad {\Delta/N}} \right)}^{j\frac{N - 1}{N}\pi \quad k\quad \Delta}}}\end{matrix}$

[0037] Mathematically, if there is no sampling frequency offset, r_(k)is equal to a_(m) ^(k)H_(k)·N, where H_(k) can be removed by afrequency-dominion equalizer 112. However, since the samplingfrequencies of the transmitter and the receiver are different (thefrequency offset is Δ), the received signal of the k-th sub-channel(k=0˜N−1) is attenuated by$\frac{\sin \left( {\pi \quad k\quad \Delta} \right)}{N\quad {\sin \left( {\pi \quad k\quad {\Delta/N}} \right)}}$

[0038] times and rotates with an angle of${\left( \frac{N - 1}{N} \right)\quad \pi \quad k\quad \Delta} \approx {\pi \quad k\quad {\Delta \quad.}}$

[0039] The former approximates to 1 and can be negligible, while thelatter is not negligible. Therefore, the sampling frequency offset Δwould cause the signals of the k-th sub channel to rotate with the angleof πkΔ for the sampled symbol.

[0040] (2) A symbol timing error is generated for subsequent sampledsymbols. Since the symbol timing error is accumulated to (N+υ)·Δ·t_(s)(here t_(s)=1/f_(s)) is after sampling a symbol. This symbol timingerror causes an additional angle rotation of${\left( \frac{N + \upsilon}{N} \right)2\quad \pi \quad k\quad \Delta} \approx {2\quad \pi \quad k\quad \Delta}$

[0041] in the k-th output of the FFT circuit 111 when sampling the nextsymbol. The approximation is tenable because υ (the length of cyclicprefix) is much smaller than N.

[0042] If the sampling frequency offset Δ is regarded as a time-variantsignal but is constant during transceiving the same symbol, in addition,the sampling frequency offset Δ is regarded as the input of theCommunication system 100, and the rotation angle of the output of theFFT circuit 111 is regarded as the output of the system 100, then thecommunication system 100 can be viewed as a sampling system, and abehavior model thereof can be derived based on the above conditions tofurther suggest the apparatus and method of the present invention.

[0043] Assume that the sampling frequency offset Δ(m) is an impulsesignal as follows:

[0044] Δ(0)=Δ

[0045] Δ(m)=0, m>0 or m<0

[0046]FIG. 4 is a diagram showing the rotation angle of the output ofthe k-th sub-channel from the FFT circuit 111. From FIG. 4, the rotatedangle θ_(k)(m) can be expressed as:

θ_(k)(m)=∠r _(k)(m)=θ₀, when m<0

=θ₀ +πkΔ, when m=0

=θ₀+2πkΔ, when m>0

[0047] where θ₀=∠a_(m) ^(k)·H_(k)

[0048] Therefore, the impulse response of the sampling system can bederived as (expressed in z-transform): $\begin{matrix}{\begin{matrix}{{\theta_{k}(z)} = {\theta_{0} + {\pi \quad k\quad {\Delta\left\lbrack {a + {bz}^{- 1} + {bz}^{- 2} + \ldots}\quad \right\rbrack}}}} \\{\approx {\theta_{0} + {\pi \quad k\quad {\Delta\left\lbrack {1 + {2z^{- 1}} + {2\quad z^{- 2}} + \ldots}\quad \right\rbrack}}}}\end{matrix}{{{{where}\quad a} = {\frac{N - 1}{N} \approx 1}},{b = {{2\left( \frac{N + \upsilon}{N} \right)} \approx 2}}}} & \left( {1\text{-}2} \right)\end{matrix}$

[0049] Since the sampling system is an approximately lineartime-invariant (LTI) system, a behavior model is established as shown inthe simplifying process of FIG. 5A to 5C. FIG. 5A is a block diagramshowing a the behavior of the sampling system, that is, how to obtainΔ(m) by choosing a pilot tone and using associated devices. In FIG. 5A,assume that the sampling frequency f_(s)(m) of the transmitter is[1+Δ_(r)(m)]f_(s) and the sampling frequency {circumflex over(f)}_(s)(m) of the receiver is [1+Δ_(r)(m)]f_(s), where both are changedaccording to the symbol number index m, and f_(s) is an ideal samplingfrequency required by the specification of the communication system 100.The dotted-line portion in FIG. 5A is the output of an estimator 501 forestimating the frequency offset Δ, and its operation is described asfollows. After selecting the pilot tone (assumed as the k-th sub-channelin FIG. 5A), divide the pilot tone by a_(m) ^(k), and pass it through anangle estimator to obtain θ_(k)(m). Next, a signal d θ_(k)(m) (i.e.2πkΔ(m)) by using a first difference device (i.e. 1−z⁻¹), and passedthrough an amplifier with a gain of $\frac{1}{2\quad \pi \quad k}$

[0050] to obtain Δ(m). It should be noted that the signal-processingdelay in FIG. 5A can be expressed as z^(−D−1), where z^(−D) is the totaldelay resulted from performing calculations (e.g. FFT) by the system100, and z₃₁ ₁ is the buffer delay, generated by the serial-to-parallelconverter (S/P) 109.

[0051]FIG. 5B is a simplified diagram of FIG. 5A. That is, if thesampling frequency offset (i.e. f_(S)(m)−{circumflex over (f)}_(s)(m))is regarded as the input of the sampling system, then θ_(k)(m) isobtained after passing the sampling frequency offset through a filterG(z) and an amplifier with a gain of πk/f_(s), where G(z) equals to the[a+bz⁻¹+bz⁻²+ . . . ] portion in Equation (1-2) multiplied by z^(−D−1),that is,

G(z)=z ^(−D−1) [a+bz ⁻¹ +bz ⁻²+ . . . ]

≈z ^(−D−1)1+2z ⁻¹ +2z ⁻²+ . . .

[0052]FIG. 5C is a further simplified diagram of FIG. 51. A behaviormodel is obtained by removing f_(s) and incorporating other devices. Inthis behavior model, the input [Δ_(t)(m)−Δ_(r)(m)] is passed through asystem response filter g(z), and an output Δ(m) is generated, where$\begin{matrix}\begin{matrix}{{g(z)} = {\frac{1}{2}{G(z)}\left( {1 - z^{- 1}} \right)}} \\{= {\frac{1}{2}{z^{{- D} - 1}\left\lbrack {a + {\left( {b - a} \right)z^{- 1}}} \right\rbrack}}} \\{\approx {\frac{1}{2}{z^{{- D} - 1}\left( {1 + z^{- 1}} \right)}}}\end{matrix} & \left( {1\text{-}3} \right)\end{matrix}$

[0053] The present invention is intended to design a tracking loop whichenables the receiver of a multi-carrier communication system tocompensate the sampling frequency offset quickly and correctly at theearly period of signal transmission. Based on the previous analysis, ifthe communication system 100 contains a device similar to the estimator501, of FIG. 5A, then the behavior model of FIG. 5C can be employed toanalyze sampling behavior of the communication system 100, that is,[Δ_(t)(m)−Δ_(r)(m)] (i.e. the sampling frequency offset) is used as aninput to a system response filter g(z), and an output Δ(m) is thengenerated. The frequency offset estimated by the estimator 501 containsa result of the frequency offset of the receiver passing through g(z),where the result is negative (i.e.−Δ_(r)(m)g(z)). Thus, if Δ_(r)(m) ispassed through a filter with a similar system response g′(z), and theoutput of the filter is added to Δ(m) for compensation, then, the sum ofthe output of the filter and Δ(m) is added to the oscillator forgenerating the sampling frequency of the receiver. The above process isrepeated and Δ_(t)(m) (and the sampling frequency of the transmitter[1+Δ_(l)(m)]f_(s)) can be tracked step by step.

[0054] It should be noted that the design of the estimator 501 is notlimited to that of FIG. 5A. For example, when the communication system100 uses RF carriers, a design of the estimator 501 different from FIG.5A needs to be adopted.

[0055]FIG. 6A is a block diagram of the apparatus for synchronizingsampling frequency and carrier frequency 600 used in the multi-carriercommunication system 100 according to the embodiment of the presentinvention. As shown in FIG. 6A, the apparatus 600 is used to synchronizesampling frequency of the receiver and that of the transmitter of thecommunication system 100, where the sampling frequency f_(s)(m) of thetransmitter is [1+Δ_(t)(m)]f_(s), the sampling frequency {circumflexover (f)}_(s)(m) of the receiver is [1+Δ_(r)(m)]f_(s), and m is thesymbol number index. The receiver includes the estimator 501 forestimating a frequency offset the apparatus 600 comprises: acompensation loop filter g′(z) 601 for generating a first output inresponse to the frequency offset compensation Δ_(r)(m) of the receiver;an adder 602 for adding the estimated frequency offset outputted fromthe estimator 501 and the first output from compensation loop filter 601to generate a second output; and a loop filter H(z) 603 for receivingthe second output from the adder 602, and for generating a new frequencyoffset compensation to replace the previous frequency offsetcompensation. The new frequency offset compensation is then fed back tothe oscillator 604 to compensate the frequency offset of the receiver.

[0056] The behavior model of FIG. 5C can be combined with the apparatus600, as shown in FIG. 6B, for further analysis. The noise n is added toreflect the practical situation. It can be derived from the compensationloop of FIG. 6B that

{[Δ_(t)(z)−Δ_(r)(z)]g(z)+n+Δ_(r)(z)g′(z)}H(z)=Δ_(r)(z)=H(z)g(z)Δ_(t)(z)+nH(z)=Δ_(t)(z){1+[g(z)−g′(z)]H(z)}  (1-4)

[0057] If we make g′(z)=g(z), then Equation (1-4) becomes

Δ_(r)(z)=H(z)g(z) Δ_(t)(z)+nH(z)_(tm ()1-5)

[0058] From Equation (1-3),${{g(z)} = {\frac{1}{2}{z^{{- D} - 1}\left( {1 + z^{- 1}} \right)}}},$

[0059] which is a low-pass filter with a finite impulse response. Theloop filter H(z) can be designed as a low-pass filter to remove thenoise, and then H(z)g(z) is also a low-pass filter. Therefore, Δ_(r)(z)in Equation (1-5) may be regarded as in outcome of the time-variantsignal Δ_(t)(z) passing through a low-pass filter. Generally speaking,Δ_(t)(z) is varied slowly and not much affected when passed through thelow-pass filter. Thus, the estimated Δ_(r)(z) would be very close toΔ_(t)(z), and the object of the present invention can then be achieved.

[0060] It should be noted that besides g(z), the compensation loopfilter g′(z) 601 may have other implementations. For instance, it isfeasible to make g′(z)=z_(−D−1) to compensate the response of g(z).Besides, H(z) can be time-variant. For example, let H(z)=1 at the earlyperiod of frequency offset tracking, and then let H(z) equal to${\frac{1}{2}\left( {1 + z^{- 1}} \right)},{\frac{1}{3}\left( {1 + z^{- 1} + z^{- 2}} \right)\quad \ldots}\quad,$

$\frac{1}{n}\left( {1 + z^{- 1} + z^{- 2} + \ldots + z^{{- n} + 1}} \right)$

[0061] in order. In this way, the frequency offset can be trackedquickly and precisely.

[0062]FIG. 7 is a flow chart of the compensation method according to theembodiment of the present invention. As shown in FIG. 7, when thereceiver receives a signal from the transmitter, the apparatus 600 ofFIG. 6A performs the following steps:

[0063]701 processing the signal by using [1+Δ_(r)(m)]f_(s) as thesampling frequency of the receiver, estimating the sampling frequencyoffset by the estimator 501, and providing Δ_(r)(m) to the compensationloop filter g′(z) 601 to obtain a first output, wherein Δ_(r)(m) is atime-variant signal;

[0064]702 adding the estimated sampling frequency offset and the firstoutput, and providing the sum to the loop filter H(z) 603 to obtain afrequency offset compensation; and

[0065]703 using the frequency offset compensation as a new Δ_(r)(m), andrepeating the previous steps.

[0066] An example is provided here to further explain the above steps.Assume there exists a rather large frequency offset, depicted as aconstant d, between the sampling frequencies of the transmitter andreceiver of the communication system 100. A typical value of d lies fromtens to one hundred ppm. This frequency offset would cause Δ(m) of FIG.6A to have an output d. However, since a delay Z_(−D−1) is generatedwhen the system 100 processes the signal, a frequency offset d would bereflected in Δ(mn) after the delay passes. Therefore, we can modify thebehavior mode of FIG. 6B to accommodate this case, as shown in FIG. 8. Atransient analysis for the communication system 100 according to themodel of FIG. 8 is performed as below:

Δ_(r)(z)=H(z)g(z)Δ_(t)(z)+d·u(z)·z ^(−D−1) H(z)  (1-6)

[0067] where u(z), the z-transform of a unit-step signal, is used losimulate the initial status of signal transceiving of the system 100.

[0068] It is assumed that Δ_(t)(z) is relatively small compared to thequantity of frequency offset d and can thus be neglected. Then Equation(1-6) becomes

Δ_(r)(z)≈d·u(z)·z ^(−D−1) H(z)

[0069] Accordingly, if there is a frequency offset d, the tracking timerequired by the method disclosed in the embodiment of the presentinsertion is only relative to H(z) and total delay Z^(−D−1), which isresulted from signal processing of the system 100. Assume${H(z)} = {\frac{1}{2}\left( {1 + z^{- 1}} \right)}$

[0070] and D=1, then

Δ_(r)(z)≈d(0.5z ⁻² +z ⁻³+ . . . )

[0071] That is, the communication system needs only a time period ofthree symbols to track the sampling frequency offset.

[0072] Furthermore, the method disclosed in the embodiment of thepresent invention can also be applied to track carrier frequency. FIG. 9is a block diagram showing that the multi-carrier communication system100 employs carriers for transmitting signals. As shown in FIG. 9, thetransmitter of the system 100 uses a mixer 901 to multiply the signalswith e^(j2πf) ^(_(c)) ^(t) , i.e. a carrier with a frequency f_(c),thereby modulating baseband signals outputted from the IFFT circuit 102to a higher frequency band for transmission, The receiver uses anothermixer 902 to demodulate received signals to the baseband by multiplyingwith e^(−j2πf) ^(_(c)) ^(t) , and then sends there into the FFT circuit111 for further demodulation. Since f_(c)≠{circumflex over (f)}_(c), itis necessary for the receiver to perform estimation and compensation ofcarrier frequency offsets. By applying the above mathematic model,similar physical Phenomena are obtained, and thus the compensationmethod of the present invention can be used to design a carrierfrequency tracking loop. Next, a mathematic analysis for the carrierfrequency offset is provided as follows:

[0073] Let f_(s) be a sampling frequency

[0074] f_(c)(m) be a carrier frequency of the transmitter

[0075] {circumflex over (f)}_(c)(m) e a carrier frequency of thereceiver

[0076] m be a symbol number index

[0077] and f_(c)(m)=f_(c)+df_(c)(m)

[0078] {circumflex over (f)}_(c)(m)=f_(c)+d{circumflex over (f)}_(c)(m)

[0079] where f_(c) is an ideal carrier frequency required by thespecification of the communication system 100.

[0080] It is reasonable to assume that the carrier frequency offset isconstant during the same symbol. Since df_(c)(m)≠d{circumflex over(f)}_(c)(m), the carrier frequency offset is generated. By inferring inthe way similar to the case of sampling frequency, the effects caused bythis frequency offset are:

[0081] (1) The output of the FFT circuit 111 generates “constellationdistortion” for the m-th symbol being sampled. If the same notations inthe previous analysis are used, then

S _(m) ^(k)′(n)=a _(m) ^(k) ·H _(k) ·e ^(jw) ^(_(k)) ^(′m) ,n=0˜N−1

[0082] where$w_{k}^{\prime} = {2\quad {\pi \left\lbrack {\frac{k}{N} + \Delta} \right\rbrack}}$

[0083] is the carrier frequency of k-th sub-channel

[0084] Δ=[df_(c)−d{circumflex over (f)}_(c)]/f_(s) is the carrierfrequency offset.

[0085] The output signal of k-th sub-channel of the FFT circuit 111 is$r_{k} = {a_{m}^{k}{H_{k}\left\lbrack \frac{\sin \quad \left( {\pi \quad N\quad \Delta} \right)}{\sin \quad \left( {\pi \quad \Delta} \right)} \right\rbrack}^{{j{({N - 1})}}\pi \quad \Delta}}$

[0086] Accordingly, if there is an offset Δ between the carrierfrequencies of the transmitter and the receiver, then the receivedsignals of all the sub-channels are attenuated by$\frac{\sin \quad \left( {\pi \quad N\quad \Delta} \right)}{N\quad \sin \quad \left( {\pi \quad \Delta} \right)}$

[0087] times and rotates with an angle of (N−1)πΔ. The formerapproximates to 1 and is negligible, while the latter is not negligible.

[0088] (2) An additional angle rotation is generated for subsequentdemodulated symbols. During the same symbol, the angle rotation resultedfrom the carrier frequency offset is accumulated as 2π(N+υ)Δ. Similarly,we can derive the behavior model of the system 100 which uses thecarrier, as shown in FIG. 10A and 10B, whereG(z) = z^(−D − 1)[a + bz⁻¹ + bz⁻² + …] ≈ z^(−D − 1)[1 + 2z⁻¹ + 2z⁻² + …]${g(z)} = {{\frac{1}{2}{z^{{- D} - 1}\left\lbrack {a + {\left( {b - a} \right)z^{- 1}}} \right\rbrack}} \approx {\frac{1}{2}{z^{{- D} - 1}\left( {1 + z^{- 1}} \right)}}}$$\quad {{a = {\frac{N - 1}{N} \approx 1}},{b = {{2 \cdot \frac{N + \upsilon}{N}} \approx 2}}}$Δ  t(m) = df_(c)(m)/f_(s), Δ  r(m) = df̂_(c)(m)/f_(s)

[0089]FIG. 11A and 11B are two other embodiments of the presentinvention. According to the above-disclosed analysis, once thecompensated frequency offset is determined, the frequency compensationcan be executed in different devices of the receiver. In FIG. 11A, thecompensated frequency offset is inputted into the mixer 902 at thereceiver via a voltage-controlled oscillator (VCO) 1101. Thevoltage-controlled oscillator 1101 is for outputting the demodulatedcarrier signals of which the frequency is adjusted according to thecompensated frequency offset. Then, the mixer 902 demodulated thesignals from the transmitter with the carrier signals. The carrierfrequency offset between the transmitter and this receiver can thus becompensated. In FIG. 11B, the compensated frequency offset is inputtedinto a phase rotator 1103 for rotating a signal outputted from the ADC107 via a phase accumulator 1102. The phase accumulator 1102 is foroutputting the phase rotated signal according to the compensatedfrequency offset. The frequency offset between the transmitter and the,receiver can thus be compensated. It should be noted that the operationof the oscillator 604 of FIG. 6A, 1101 of FIG. 11A, and 1102 of FIG. 11Bmay be different and the signal outputted from the oscillator 604 ofFIG. 6A, 110 of FIG. 11A, and 1102 of FIG. 11B may be in differentformat, as shown in FIG. 6A, FIG. 11A, and FIG. 11B respectively.However, people who skilled in the art can easily accomplish theembodiments disclosed in FIG. 11A and FIG. 11B according to thedisclosure of the specification.

[0090] While the present invention has been shown and described withreference to two preferred embodiments thereof, and in terms of theillustrative drawings, it should be not considered as limited thereby.Various possible modification, omission, and a lterations could beconceived of by one skilled in the art to the form and the content ofany particular embodiment, without departing from the scope and thespirit of the present invention.

What is claimed is:
 1. An apparatus for synchronizing frequencies of areceiver and a transmitter of a multi-carrier communication system,wherein the receiver includes an estimator for estimating a frequencyoffset according to a signal received from the transmitter and producingan estimated frequency offset, comprising: a first loop filter,generating a first output in response to a frequency offsetcompensation; an adder, adding the estimated frequency offset and thefirst output and generating a second output; and a second loop filter,generating the frequency offset compensation according to the secondoutput.
 2. The apparatus of claim 1, wherein the frequency offset is asampling frequency offset between the transmitter and the receiver. 3.The apparatus of claim 1, wherein the frequency offset is a carriersignal offset between the receiver and the transmitter.
 4. The apparatusof claim 1, further comprising: an oscillator coupled to the second loopfilter and generating a compensated signal of the receiver according tothe frequency offset compensation.
 5. The apparatus of claim 1, furthercomprising: an oscillator coupled to the second loop filter andgenerating a sampling frequency of the receiver according to thefrequency offset compensation. an analog-to-digital converter forsampling a signal from the transmitter with the sampling frequency togenerate a digital received signal; and a transformer for generating thereceived signal according, to the digital received signal.
 6. Theapparatus of claim 1, further comprising: a phase accumulator foroutputting a phase rotated control signal according to the frequencyoffset compensation; a phase rotator for rotating a signal from thetransmitter to generate a phase rotated signal according to the phaserotated control signal; and a transformer for generating the receivedsignal according to the phase rotated signal.
 7. The apparatus of claim1, further comprising: an oscillator for outputting a carrier signalaccording to the frequency offset compensation; a mixer for mixing asignal from the transmitter with the carrier signal, and outputting amixed signal; and a transformer for generating the received signalaccording to the mixed signal.
 8. The apparatus of claim 1, wherein thesecond loop filter is a time-variant low-pass filter.
 9. The apparatusof claim 1, wherein the second loop filter is a time-invariant low-passfilter.
 10. A method for synchronizing frequencies of a receiver and atransmitter of a multi-carrier communication system, comprising:estimating a frequency offset, wherein the frequency offset is thedifference between a frequency of the transmitter and t frequency of thereceiver; generating a frequency offset compensation according to thefrequency offset; generating a first output by a first loop-filteraccording to the frequency offset compensation; adding the frequencyoffset and the first output; and generating a second output as thefrequency offset compensation by a second loop-filter according to aresult of the adding.
 11. The method of claim 10, wherein the frequencyoffset is a sampling frequency offset between the transmitter and thereceiver.
 12. The method of claim 10, further comprising: modifying asampling frequency of the receiver according to the frequency offsetcompensation.
 13. The method of claim 10, further comprising: modifyinga phase rotated control signal of the receiver according to thefrequency offset compensation.
 14. The method of claim 10, furthercomprising: modifying a carrier signal of the receiver according to hefrequency offset compensation.
 15. An apparatus for synchronizingfrequencies of a receiver and a transmitter of a multi-carriercommunication system, comprising: a frequency estimator for producing anestimated frequency offset according to a received signal; a first loopfilter, generating a first output in response to a frequency offsetcompensation; an adder, adding the estimated frequency offset and thefirst output and generating a second output; and a second loop filter,generating the frequency offset compensation according to the secondoutput.
 16. The apparatus of claim 15, further comprising: an oscillatorfor outputting a sampling frequency according to the frequency offsetcompensation; an analog-to-digital converter for sampling a signal fromthe transmitter with the sampling frequency to generate a digitalreceived signal; and a transformer for generating the received signalaccording to the digital received signal.
 17. The apparatus of claim 15,further comprising: a phase accumulator for outputting a phase rotatedcontrol signal according to the frequency offset compensation; a phaserotator for rotating a signal from the transmitter to generate a phaserotated signal according to the phase rotated control signal; and atransformer for generating the received signal according to the phaserotated signal.
 18. The apparatus of claim 15, further comprising: anoscillator for outputting a carrier signal according to the frequencyoffset compensation; a mixer for mixing a signal from the transmitterwith the carrier signal, and outputting a mixed signal; and atransformer for generating the received signal according to the mixedsignal.
 19. The apparatus of claim 15, wherein the second loop filter isa time-variant low-pass filter.
 20. The apparatus of claim 15, whereinthe second loop filter is a time-invariant low-pass filter.